An adaptive feedback control for chaos synchronization of nonlinear systems with different order

This paper deals with the chaotic synchronization of fourth-order system and second driven oscillators. Such a problem is related to the synchronization of strictly different chaotic systems. We show that the dynamical evolution of second-order driven oscillators can be synchronized with the projection on canonical planes of a fourth-order chaotic system. In this sense, it is said that the synchronization is achieved in reduced-order. Based on the Lyapunov approach, the adaptation law is determined to tune the controller gain vector in order to track a predetermined linearizing feedback control. An application to secure chaotic communication is also discussed. Numerical simulations are presented to demonstrate the efficiency of the proposed synchronization and secure communication schemes.     

Reduced-order synchronization of uncertain chaotic systems via adaptive control

We consider the coupling of two uncertain dynamical systems with different orders using an adaptive feedback linearization controller to achieve reduced-order synchronization between the two systems. Reduced-order synchronization is the problem of synchronization of a slave system with projection of a master system. The synchronization scheme is an exponential linearizing-like controller and a state/uncertainty estimator. As an illustrative example, we show that the dynamical evolution of a second-order driven oscillator can be synchronized with the canonical projection of a fourth-order chaotic system. Simulation results indicated that the proposed control scheme can significantly improve the synchronousness performance. These promising results justify the usefulness of the proposed output feedback controller in the application of secure communication.     

Synchronization in reduced-order of chaotic systems via control approaches based on high-order sliding-mode observer

The reduced-order synchronization problem of two chaotic systems (master–slave) with different dimension and relative degree is considered. A control scheme based on a high-order sliding-mode observer-identifier and a feedback state controller is proposed, where the trajectories of slave can be synchronized with a canonical projection of the master. Thus, the reduced-order synchronization is achieved in spite of master/slave mismatches. Simulation results are provided in order to illustrate the performance of the proposed synchronization scheme.     

Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control

In this paper, we investigate the synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. Using a combination of Riccati differential equation approach, Lyapunov-Krasovskii functional, inequality techniques, some sufficient conditions for exponentially stability of the error system are formulated in form of a solution to the standard Riccati differential equation. The designed controller ensures that the synchronization of non-autonomous chaotic systems are proposed via delayed feedback control and intermittent linear state delayed feedback control. Numerical simulations are presented to illustrate the effectiveness of these synchronization criteria.     

受扰不确定混沌系统的降阶修正函数投影同步

研究了具有不同阶数的受扰不确定混沌系统的降阶修正函数投影同步问题.基于Lyapunov稳定性理论和自适应控制方法,设计了统一的非线性状态反馈控制器和参数更新规则,使得混沌响应系统按照相应的函数尺度因子矩阵和混沌驱动系统的部分状态变量实现同步.方法考虑了实际系统中的模型不确定性和外界扰动,具有较强的实用性和鲁棒性.数值仿真证明了控制方法的有效性.     

Synchronization of non-identical chaotic delayed fuzzy cellular neural networks based on sliding mode control

-In this paper, we investigate the synchronization problems of chaotic fuzzy cellular neural networks with time-varying delays. To overcome the difficulty that complete synchronization between non-identical chaotic neural networks cannot be achieved only by utilizing output feedback control, we use a sliding mode control approach to study the synchronization of non-identical chaotic fuzzy cellular neural networks with time-varying delays, where the parameters and activation functions are mismatched. This research demonstrates the effectiveness of application in secure communication. Numerical simulations are carried out to illustrate the main results.     

Robust synchronization of chaotic non-autonomous systems using adaptive-feedback control

In this paper, we apply the simple adaptive-feedback control scheme to synchronize a class of chaotic non-autonomous systems. Based on the invariance principle of differential equations, some generic sufficient conditions for global asymptotic synchronization are obtained. Unlike the usual linear feedback, the variable feedback strength is automatically adapted to completely synchronize two identical systems and simple to implement in practice. As illustrative examples, synchronization of two parametrically excited chaotic pendulums and that of two 4D new systems are considered here. Numerical simulations show the proposed method is effective and robust against the effect of noise.     

Some simple global synchronization criterions for coupled time-varying chaotic systems

Based on the Lyapunov stabilization theory and matrix measure, this paper proposes some simple generic criterions of global chaos synchronization between two coupled time-varying chaotic systems from a unidirectional linear error feedback coupling approach. These simple criterions are applicable to some typical chaotic systems with different types of nonlinearity, such as the original Chua’s circuit and the Rössler chaotic system. The coupling parameters are determined according to the new criterion so as to ensure the coupled systems’ global chaos synchronization.     

Simple adaptive variable structure control for unknown chaotic systems

This paper proposes two novel adaptive variable structure tracking controllers for a large class of chaotic systems with unknown dynamics in presence of both external disturbances and input nonlinearities. The pros and cons of each proposed methodology is also represented. In order to eliminate the chattering effect in the former controlled system, two corresponding fuzzy adaptive controllers are presented. Besides, synchronization of two non-identical uncertain chaotic systems is investigated using our proposed methods in both full and reduced-order forms. It can be seen that not only our proposed control schemes can be applied to a wide class of uncertain chaotic systems but also it is simple to implement in practical application. Finally, the proposed methods are applied to some famous chaotic systems to verify the effectiveness of the proposed methods.     

Global asymptotical synchronization of chaotic neural networks by output feedback impulsive control: An LMI approach

In this paper, impulsive control for master–slave synchronization schemes consisting of identical chaotic neural networks is studied. Impulsive control laws are derived based on linear static output feedback. A sufficient condition for global asymptotic synchronization of master–slave chaotic neural networks via output feedback impulsive control is established, in which synchronization is proven in terms of the synchronization errors between the full state vectors. An LMI-based approach for designing linear static output feedback impulsive control laws to globally asymptotically synchronize chaotic neural networks is discussed. With the help of LMI solvers, linear output feedback impulsive controllers can be easily obtained along with the bounds of the impulsive intervals for global asymptotic synchronization. The method is finally illustrated by numerical simulations.     

Observer-based chaotic synchronization in the presence of unknown inputs

This paper deals with the problem of synchronization of chaotic dynamical systems. We consider a drive-response type of synchronization via a scalar transmitted signal. Unlike most works we consider the presence of some unknown inputs in the drive system and that no knowledge about their nature is available. A reduced-order observer-based response system is designed to synchronize with the missing states. We show that under some assumptions the synchronization is exponentially achieved. The efficiency of our method is confirmed by numerical simulations of two well-known chaotic systems: Chua’s circuit and Lur’e system.     

Adaptive synchronization of Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays

In this paper, we investigate the synchronization problem of chaotic Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays. An adaptive linear feedback controller is designed to guarantee that the response system can be synchronized with a drive system by utilizing Lyapunov stability theory and parameter identification. Our synchronization criteria are easily verified and do not need to solve any linear matrix inequality. These results generalize a few previous known results and remove some restrictions on amplification function and time delay. This research also demonstrates the effectiveness of application in secure communication. Numerical simulations are carried out to illustrate the main results.     

Synchronization of chaos in non-identical parametrically excited systems

In this paper, we investigate the synchronization of chaotic systems consisting of non-identical parametrically excited oscillators. The active control technique is employed to design control functions based on Lyapunov stability theory and Routh–Hurwitz criteria so as to achieve global chaos synchronization between a parametrically excited gyroscope and each of the parametrically excited pendulum and Duffing oscillator. Numerical simulations are implemented to verify the results.     

Adaptive cluster synchronization in networks with time-varying and distributed coupling delays

This paper studies the adaptive cluster synchronization of a generalized linearly coupled network with time-varying delay and distributed delays. This network includes nonidentical nodes displaying different local dynamical behaviors, while for each cluster of that network the internal dynamics is uniform (such as chaotic, periodic, or stable behavior). In particular, the generalized coupling matrix of this network can be asymmetric and weighted. Two different adaptive laws of time-varying coupling strength and a linear feedback control are designed to achieve the cluster synchronization of this network. Some sufficient conditions to ensure the cluster synchronization are obtained by using the invariant principle of functional differential equations and linear matrix inequality (LMI). Numerical simulations verify the efficiency of our proposed adaptive control method.     

Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication

This paper investigates drive-response synchronization of chaotic systems with discontinuous right-hand side. Firstly, a general model is proposed to describe most of known discontinuous chaotic system with or without time-varying delay. An uniform impulsive controller with multiple unknown time-varying delays is designed such that the response system can be globally exponentially synchronized with the drive system. By utilizing a new lemma on impulsive differential inequality and the Lyapunov functional method, several synchronization criteria are obtained through rigorous mathematical proofs. Results of this paper are universal and can be applied to continuous chaotic systems. Moreover, numerical examples including discontinuous chaotic Chen system, memristor-based Chua’s circuit, and neural networks with discontinuous activations are given to verify the effectiveness of the theoretical results. Application of the obtained results to secure communication is also demonstrated in this paper.     

Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems

In this paper, a new type of anticipating synchronization, called time-varying anticipating synchronization, is defined firstly. Then novel adaptive schemes for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems are designed based on the Lyapunov function and invariance principle. The update gain of coupling strength can be automatically adapted to a suitable strength depending on the initial values and can be properly chosen to adjust the speed of achieving synchronization, so these schemes are analytical and simple to implement in practice. A classical chaotic dynamical system is used to demonstrate the effectiveness of the proposed adaptive schemes with or without parameter uncertainties.     

Adaptive control and synchronization of chaotic systems consisting of Van der Pol oscillators coupled to linear oscillators

This paper deals with the problem of control and synchronization of coupled second-order oscillators showing a chaotic behavior. A classical feedback controller is first used to stabilize the system at its equilibrium. An adaptive observer is then designed to synchronize the states of the master and slave oscillators using a single scalar signal corresponding to an observable state variable of the driving oscillator. An interesting feature of the proposed approach is that it can be used for chaos control as well as synchronization purposes. Numerical simulations results confirming the analytical predictions are shown and pspice simulations are also performed to confirm the efficiency of the proposed control scheme.     

Global synchronization criterion and adaptive synchronization for new chaotic system

This paper proposes two schemes of synchronization of two four-scorll chaotic attractor, a simple global synchronization and adaptive synchronization in the presence of unknown system parameters. Based on Lyapunov stability theory and matrix measure, a simple generic criterion is derived for global synchronization of four-scorll chaotic attractor system with a unidirectional linear error feedback coupling. This methods are applicable to a large class of chaotic systems where only a few algebraic inequalities are involved. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization method.     

Synchronization of neural networks based on parameter identification and via output or state coupling

For neural networks with all the parameters unknown, we focus on the global robust synchronization between two coupled neural networks with time-varying delay that are linearly and unidirectionally coupled. First, we use Lyapunov functionals to establish general theoretical conditions for designing the coupling matrix. Neither symmetry nor negative (positive) definiteness of the coupling matrix are required; under less restrictive conditions, the two coupled chaotic neural networks can achieve global robust synchronization regardless of their initial states. Second, by employing the invariance principle of functional differential equations, a simple, analytical, and rigorous adaptive feedback scheme is proposed for the robust synchronization of almost all kinds of coupled neural networks with time-varying delay based on the parameter identification of uncertain delayed neural networks. Finally, numerical simulations validate the effectiveness and feasibility of the proposed technique.     

Adaptive synchronization to a general non-autonomous chaotic system and its applications

In this paper, new adaptive synchronous criteria for a general class of n-dimensional non-autonomous chaotic systems with linear and nonlinear feedback controllers are derived. By suitable separation between linear and nonlinear terms of the chaotic system, the phenomenon of stable chaotic synchronization can be achieved using an appropriate adaptive controller of feedback signals. This method can also be generalized to a form for chaotic synchronization or hyper-chaotic synchronization. Based on stability theory on non-autonomous chaotic systems, some simple yet less conservative criteria for global asymptotic synchronization of the autonomous and non-autonomous chaotic systems are derived analytically. Furthermore, the results are applied to some typical chaotic systems such as the Duffing oscillators and the unified chaotic systems, and the numerical simulations are given to verify and also visualize the theoretical results.